[FOM] Physical Theories and Hypercomputation

[Antonino Drago]

If  quantum mechanics is considered necessary , as Vaughan Pratt does, for 
evaluating the "Hypercomputability", one cannot forget that the logic of 
quantum mechanics is surely a non-classical one; and since constructive 
mathematics is usually considered a mathematics under intuitionist logic, it 
would be important to focus the attention on which mathematical questions 
are not treatable by contructive mathematics.
This apoproach reverses the usual one: it is constructive mathematics the 
basic, whereas the classical mathematics has to justify its introduction.
One more important reference on the subject is S. Feferman's specific 
chapter (in "In the Light of Logic", Oxford, 1998) about  the adequacy of 
Weyl's mathematics (a more powerful version of constructive mathematics) to 
quantum mechanics.
Best regards
Antonino Drago

----- Original Message ----- 
From: "Dmytro Taranovsky" <dmytro at mit.edu>
To: <fom at cs.nyu.edu>
Sent: Sunday, March 11, 2012 2:02 PM
Subject: Re: [FOM] Physical Theories and Hypercomputation


> On 03/10/2012 11:12 AM, Vaughan Pratt wrote:
>> In response to Dmytro Taranovsky I would say simply that any speculation 
>> about the computational capability of nature that ignores the 
>> implications of quantum mechanics is dead on arrival.
> I was writing in general terms that are applicable to both classical and 
> quantum physics.  In both cases, key to recursiveness is the finite and 
> approximate nature of observations (which in quantum mechanics is 
> reinforced by the uncertainty principle).  In both cases, singularities 
> form a potential loophole.
>
> In quantum mechanics, under the multiverse interpretation, time evolution 
> is exact and deterministic, like in a classical system.  However, to 
> relate the theory to our experience, the theory is commonly presented in 
> terms of a classical observer interacting with a quantum system.  Some 
> states that make sense classically, such as a particle with an exact 
> position and exact momentum, do not exist in a quantum system, which is 
> where the uncertainty principle comes from.
>
> For some classical theories, a source of hypercomputation is a computer 
> with infinitely-self-shrinking parts, which is ruled out by ordinary 
> quantum physics.  However, some quantum theories have their own issues (as 
> in potential divergences), such as contribution of states with arbitrarily 
> high energy to the result.  For quantum field theories, convergence is a 
> major open question, and renormalizability only partially addresses it. 
> If the absence of divergences is proved (which is far from certain), then 
> one might be able to show that (ignoring potential nonrecursiveness of 
> physical constants) under appropriate assumptions, effective time 
> evolution under the Standard Model is in BQP.
>
> Vaughan Pratt also makes an intuitive argument about limited precision of 
> observations.  One correction is that 34 digits of precision is for an 
> observer with about 1 Joule of energy and 1 second of time; potential 
> precision increases linearly with E*t (and hence the number of digits of 
> precision increases logarithmically).  An interesting problem would be to 
> formalize the argument and to prove -- or refute -- that for certain 
> quantum theories, the fine structure constant (or some other constant) 
> cannot be computed to 100 decimal places in a reasonable amount of time. 
> One question here is how sensitive a many-particle system can be to the 
> precise value of the constants.
>
> Sincerely,
> Dmytro Taranovsky
>
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